Automorphisms of Right-Angled Coxeter Groups

Abstract

If (𝑊,𝑆) is a right-angled Coxeter system, then Aut(𝑊) is a semidirect product of the group Aut^∘(𝑊) of symmetric automorphisms by the automorphism group of a certain groupoid. We show that, under mild conditions, Aut^∘(𝑊) is a semidirect product of Inn(𝑊) by the quotient Out^∘(𝑊)=Aut^∘(𝑊)/Inn(𝑊). We also give sufficient conditions for the compatibility of the two semidirect products. When this occurs there is an induced splitting of the sequence 1→Inn(𝑊)→Aut(𝑊)→Out(𝑊)→1 and consequently, all group extensions 1→𝑊→𝐺→𝑄→1 are trivial.

1. Introduction

A Coxeter group 𝑊 is determined by its diagram Γ. It is known that in certain cases, 𝑊 determines Γ as well (see, e.g., [1, 2]). This is the case for right-angled Coxeter groups [3, 4], where the only relations are 𝑠²=1 for all generators 𝑠 and 𝑠𝑡=𝑡𝑠 for some pairs of generators 𝑠 and 𝑡. For right-angled Coxeter groups, it is convenient to consider the Coxeter diagram (rather than the classical Coxeter graph): the presence of an edge with endpoints 𝑠 and 𝑡 means that 𝑠 and 𝑡 commute in 𝑊.

The properties of a right-angled Coxeter group 𝑊 depend almost exclusively on the combinatorics of the diagram Γ. This is especially evident in the study of Aut(𝑊). For example, the groupoid 𝔉(Γ) consisting of the vertex sets of complete subgraphs of Γ plays an important role in [5] where Tits exhibits a split exact sequence1→Aut^∘(𝑊)→Aut(𝑊)→Aut(𝔉(Γ))→1. (1.1) Tits also defines 𝑊 to have propriété I if the complementary graph Γ^𝐶 has no triangles. He goes on to show that if 𝑊 has propriété I, then Aut^∘(𝑊) is isomorphic to Inn(𝑊).

In the present article, we focus on the set 𝔐(Γ) whose members are the vertex sets of maximal complete subgraphs. We say that Γ has condition C if there exist ℤ∈𝔐(Γ) and a collection 𝑇₁,…,𝑇_𝑛∈𝔐(Γ)−{𝑍} such that (C1)𝑇_𝑖∩𝑍≠∅ for 1≤𝑖≤𝑛,(C2) for each 𝑣∈𝑍, the cardinality of the set {𝑖∣𝑣∉𝑇_𝑖} is odd. When Γ has condition C, the subgraph spanned by 𝑍∪𝑇₁∪⋯∪𝑇_𝑛 has propriété I. Thus, our condition C is a “local version” of Tits' propriété I.

Motivated by the results of Tits regarding sequence (1.1), we consider the sequence1→Inn(𝑊)→Aut(𝑊)→Out(𝑊)→1. (1.2)A splitting of this sequence implies that all extensions of the form1→𝑊→𝐺→𝑄→1 (1.3)are trivial (cf. [6]). Clearly (1.2) does not always split. For example, ℤ₂ is a right-angled Coxeter group and0→ℤ₂→ℤ₄→ℤ₂→0 (1.4)is a nontrivial extension. On the other hand, if 𝑊 has no center, then finding nontrivial extensions of 𝑊 is surprisingly difficult. Whether (1.2) splits for all 𝑊 with trivial center is currently an open question.

The group Aut^∘(𝑊) has been studied extensively in [5, 7] and is called the group of symmetric automorphisms of 𝑊. We approach the problem of whether (1.2) splits by considering the following.

(a) Does the sequence1→Inn(𝑊)→Aut^∘(𝑊)→Aut^∘(𝑊)/Inn(𝑊)→1 (1.5)split?(b) Are the splittings of (1.1) and (1.5) compatible? A positive answer to both (a) and (b) implies that (1.2) is a split extension. We show in Theorem 5.4 that if Γ has condition C, then (1.5) splits. To obtain a splitting of (1.2), we show that the action of Aut(𝔉(Γ)) on Aut^∘(𝑊) is compatible if Γ is not “too symmetrical.” More precisely, our main result is the following.

Theorem 1.1. If Γ has condition C and each 𝛼∈Aut(𝔉(Γ)) leaves all vertices of 𝑍 invariant, then (1.2) is a split exact sequence.

It was recently shown [8] that (1.5) always splits. However, this result does not lead to a generalization of Theorem 1.1 as the splitting given there is not, in general, compatible with (1.1).

2. Right-Angled Coxeter Groups

Coxeter groups are typically defined by presentations, and there are various conventions for representing such presentations diagramatically. In this section, we review some definitions and important properties, focusing exclusively on the right-angled case. See [9] or [10] for a comprehensive treatment.

If 𝑋 is any set, let 𝑃₂(𝑋) denote the set of subsets of 𝑋 with cardinality 2. Definition 2.1. Given a finite set 𝑆 and 𝐸⊆𝑃₂(𝑆), let Γ=(𝑆,𝐸) denote the undirected graph with vertex set 𝑆 and edge set 𝐸 (note that Γ does not have loops or parallel edges). As such graphs are often used to represent right-angled Coxeter groups, Γ is called a Coxeter diagram.Definition 2.2. Given 𝑇⊆𝑆, set 𝐸_𝑇=𝐸∩𝑃₂(𝑇). The graph Γ_𝑇=(𝑇,𝐸_𝑇) is the subgraph of Γspanned by 𝑇. A complete subgraph is maximal if it is not properly contained in any complete subgraph of Γ.Definition 2.3. The presentation𝒫(Γ)=⟨𝑆∣𝑠²,(𝑡𝑢)²;𝑠∈𝑆,{𝑡,𝑢}∈𝐸⟩ (2.1)is the Coxeter presentation defined byΓ.Definition 2.4. A group 𝑊 is a right-angled Coxeter group if it has a presentation 𝒫(Γ) defined by some Coxeter diagram Γ. In this case, one writes  𝑊=𝑊(Γ) and calls 𝑊 the right-angled Coxeter group defined by Γ. The pair (𝑊,𝑆) is a right-angled Coxeter system.Remark 2.5. The Coxeter diagram is not the same as the traditional Dynkin diagram. Indeed, as graphs, the Coxeter and Dynkin diagrams are complementary.

Clearly, each Coxeter diagram defines a unique right-angled Coxeter group. On the other hand, to recover the diagram from a group one must first choose a particular Coxeter presentation. It is natural to wonder whether nonisomorphic diagrams might define isomorphic groups. The relationship between right-angled Coxeter groups and their diagrams is clarified by the following result.

Theorem 2.6 (Radcliffe [3]). If (𝑊,𝑆) and (𝑊,𝑆^′) are right-angled Coxeter systems for 𝑊, then there is an automorphism 𝜌∶𝑊→𝑊 such that 𝜌(𝑆)=𝑆^′.

A similar result was also obtained by Castella in [4]. Definition 2.7. A subgroup of 𝑊 generated by a subset of 𝑆 is called a special subgroup. If 𝑇⊆𝑆, it is customary to write 𝑊_𝑇 for the subgroup generated by 𝑇. Finite special subgroups are called spherical subgroups. A subgroup of 𝑊 is parabolic if it is conjugate to a special subgroup.

We conclude this section with statements of some of the remarkable properties enjoyed by special subgroups. For proofs, see [9] or [10]. Theorem 2.8. If 𝐴,𝐵⊂𝑆, then 𝑊_𝐴∩𝑊_𝐵=𝑊_𝐴∩𝐵.Theorem 2.9. If 𝑇⊆𝑆, then 𝑊_𝑇 is the right-angled Coxeter group defined by Γ_𝑇.Corollary 2.10. The following are equivalent:
(a)𝑊_𝑇 is spherical;(b)𝑊_𝑇 is an elementary abelian 2-group of rank |𝑇|;(c)Γ_𝑇 is complete.

3. Automorphisms of Right-angled Coxeter Groups

For the remainder of this article, let 𝑊=𝑊(Γ) be the right-angled Coxeter group defined by the connected Coxeter diagram Γ=(𝑆,𝐸). It is easily verified that, under the operation of symmetric difference, the set𝔉(Γ)={𝑇⊆𝑆∣𝑊_𝑇isfinite} (3.1)is a commutative groupoid with identity ∅.

In [5], Tits uses the group of automorphisms of 𝔉(Γ) to exhibit Aut(𝑊) as a semidirect product. We sketch the construction. Let 𝜎∈Aut(𝑊). It is well known that every maximal finite subgroup of 𝑊 is parabolic (see, e.g., [11, Lemma 3.2.1]). Consequently, every finite subgroup of 𝑊 is conjugate into a spherical subgroup. It follows that, for each 𝑇∈𝔉(Γ), there is a unique minimal 𝑇∈𝔉(Γ) such that 𝜎(𝑊_𝑇) is conjugate into 𝑊_𝑇. The map𝑞∶Aut(𝑊)→Aut(𝔉(Γ)) (3.2)given by 𝑞(𝜎)(𝑇)=𝑇 is an epimorphism. Definition 3.1. Let Aut^∘(𝑊) be the kernel of 𝑞. Elements of Aut^∘(𝑊) are called symmetric automorphisms of 𝑊.

Given 𝛼∈Aut(𝔉(Γ)), consider 𝛼∈Aut(𝑊) defined by𝛼(𝑠)=_{𝑡∈𝛼(𝑠)}𝑡 (3.3)for all 𝑠∈𝑆. A main result of [5] is the following theorem. Theorem 3.2. The mapping Aut(𝔉(Γ))→Aut(𝑊) given by 𝛼↦𝛼 is a splitting of the sequence1→Aut^∘(𝑊)→Aut(𝑊)→Aut(𝔉(Γ))→1. (3.4)

Let 𝑑_Γ be the standard path metric on Γ that assigns length one to each edge. For each 𝑠∈𝑆, define𝑠^∗={𝑡∈𝑆∣𝑑_Γ𝑠(𝑠,𝑡)≤1},_⟂={𝑡∈𝑆∣𝑑_Γ(𝑠,𝑡)≥2}. (3.5) The subgraph Γ_𝑠⟂ spanned by the elements of 𝑠^⟂ gives rise to certain generators of Aut^∘(𝑊) as follows. If 𝐾 is the vertex set of a connected component of Γ_𝑠⟂, then the map 𝜎∶𝑆→𝑊 given by𝜎(𝑡)=𝑠𝑡𝑠,𝑡∈𝐾,𝑡,𝑡∉𝐾, (3.6)extends to a unique involution 𝜎_𝑠𝐾∈Aut^∘(𝑊). The following is easily deduced from [7]. Theorem 3.3. For each 𝑠∈𝑆, let 𝐾^𝑠₁,…,𝐾^𝑠_𝑚𝑠 be the vertex sets of the components of Γ_𝑠⟂. Then Aut^∘(𝑊) is generated by the set {𝜎_𝑠𝐾^𝑠_𝑖∣𝑠∈𝑆,1≤𝑖≤𝑚_𝑠}.Remark 3.4. In [7], Mühlherr gives a complete presentation for Aut^∘(𝑊) based on a slightly different set of generators.

4. Symmetric Automorphisms

The subsets of 𝑆 that generate maximal spherical subgroups of 𝑊 play a key role in the subsequent development. As such, we define𝔐(Γ)={𝑇⊆𝑆∣𝑊_𝑇isamaximalfinitesubgroup}. (4.1)Note that 𝔐(Γ) is in one-to-one correspondence with the family of maximal complete subgraphs of Γ. The global behavior of a symmetric automorphism is governed by the following observation: if 𝜙∈Aut^∘(𝑊) and 𝑇∈𝔐(Γ), then there exists an element 𝑎_𝑇=𝑎_𝑇(𝜙)∈𝑊 such that 𝜙(𝑥)=𝑎_𝑇𝑥𝑎_𝑇⁻¹ for all 𝑥∈𝑊_𝑇.Remark 4.1. When 𝑇∈𝔐(Γ),𝑊_𝑇 is its own centralizer. Thus, the element 𝑎_𝑇 above is determined up to right multiplication by any member of 𝑊_𝑇 (i.e., up to choice of representative for the coset 𝑎_𝑇𝑊_𝑇).

If 𝑇,𝑈∈𝔐(Γ) and 𝑇∩𝑈≠∅, then 𝑎_𝑇𝑥𝑎_𝑇⁻¹=𝑎_𝑈𝑥𝑎_𝑈⁻¹ for all 𝑥∈𝑊_𝑇∩𝑊_𝑈=𝑊_𝑇∩𝑈. Consequently, 𝑎_𝑇⁻¹𝑎_𝑈 lies in the centralizer of 𝑊_𝑇∩𝑈.Definition 4.2. With 𝜙,𝑇,𝑈 as above, let 𝛾_𝑇𝑈=𝛾_𝑇𝑈(𝜙)=𝑎_𝑇⁻¹𝑎_𝑈. A representative of the double coset 𝑊_𝑇𝛾_𝑇𝑈𝑊_𝑈 is called a 𝜙-transition from 𝑊_𝑇 to 𝑊_𝑈.Remark 4.3. If 𝑇,𝑈,𝑉∈𝔐(Γ) have pairwise nonempty intersection, then 𝛾_𝑇𝑇=1,𝛾_𝑉𝑈=(𝛾_𝑈𝑉)⁻¹, and 𝛾_𝑇𝑈𝛾_𝑈𝑉=𝛾_𝑇𝑉. As the terminology suggests, the transitions are in some sense a group-theoretic analogue of the change-of-coordinate maps on a manifold.

For the remainder of this article we fix an element 𝑍∈𝔐(Γ). For much of what follows, 𝑍 may be chosen arbitrarily; in Section 5 we show that a preferred choice may exist.

Given 𝑥∈𝑊, let 𝜓_𝑥 be the inner automorphism of 𝑊 given by 𝑤↦𝑥𝑤𝑥⁻¹ for all 𝑤∈𝑊. Restricting our attention to 𝑍 we defineInn(𝑍)={𝜓_𝑠∈Inn(𝑊)∣𝑠∈𝑊_𝑍},Fix^∘(𝑍)={𝜙∈Aut^∘(𝑊)∣𝜙(𝑥)=𝑥∀𝑥∈𝑍}. (4.2) In other words, Fix^∘(𝑍) is the pointwise stabilizer of 𝑊_𝑍 under the action of Aut^∘(𝑊) on W. Observe that, since the subgraph Γ_𝑍 spanned by 𝑍 is complete, Inn(𝑍) is Abelian.

Clearly, Inn(𝑍) is a subgroup of Fix^∘(𝑍). Let 𝜋 be the restriction to Fix^∘(𝑍) of the natural map Aut^∘(𝑊)→Out^∘(𝑊) and choose a class [𝜙]∈Out^∘(𝑊) with representative 𝜙∈Aut^∘(𝑊). If 𝑓 is the restriction of 𝜙 to 𝑊_𝑍, then 𝑓(𝑥)=𝑎_𝑍𝑥𝑎_𝑍⁻¹ for all 𝑥∈𝑊_𝑍. Since [𝑓⁻¹𝜙]=[𝜙] and 𝑓⁻¹𝜙∈Fix^∘(𝑍), it follows that 𝜋 is onto. We have established the following. Theorem 4.4. The sequence0→Inn(𝑍)−−→^𝑖Fix^∘(𝑍)−−−^𝜋→Out^∘(𝑊)→1 (4.3)is a central extension.

In Section 5, we construct a retraction of the mapping Inn(𝑍)→−^𝑖Fix^∘(𝑍). We now describe the key ingredient used in this construction. For each 𝑇∈𝔉(Γ), let𝐶(𝑇)=_𝑣∈𝑇𝑣^∗. (4.4)It follows at once that 𝑊_{𝐶(𝑇∩𝑍)} is the centralizer of 𝑊_𝑇∩𝑍 in 𝑊 (cf. [5, page 350]). The function𝜋_𝑇∶𝐶(𝑇∩𝑍)→𝑍−𝑇 (4.5)given by𝜋_𝑇(𝑠)=1,𝑠∉𝑍−𝑇,𝑠,𝑠∈𝑍−𝑇, (4.6)extends uniquely to a retraction 𝑊_{𝐶(𝑇∩𝑍)}→𝑊_𝑍−𝑇, which we also denote by 𝜋_𝑇.

5. Splittings

As in Section 4, we assume that 𝑍 is a fixed element of 𝔐(Γ).Definition 5.1. One says that 𝑍 satisfies condition C if there exist elements 𝑇₁,𝑇₂,…,𝑇_𝑛∈𝔐(Γ)−{𝑍} such that the following conditions hold.
(C1)𝑇_𝑖∩𝑍≠∅ for each 1≤𝑖≤𝑛.(C2) For each 𝑣∈𝑍, the cardinality of the set {𝑖∣𝑣∉𝑇_𝑖} is odd. If Γ contains a maximal complete subgraph Ω whose vertex set 𝑍 satisfies condition C, then one says that Γ has condition C. When Γ has condition C, then for each 1≤𝑖≤𝑛 above let Δ_𝑖 denote the maximal complete subgraph of Γ spanned by 𝑇_𝑖.Note that, since Γ is connected, when 𝑛=0 our hypotheses imply that Γ is complete. In this case, 𝑊 is Abelian and the results below are trivial. Thus, we may assume that 𝑛 is a positive integer. Example 5.2. We illustrate condition C with some examples.
(a) If Γ contains a maximal complete subgraph Ω with an even number of vertices each of which meets exactly one other complete subgraph of Γ, then Γ has condition C. For example, any Coxeter diagram Γ that contains Figure 1(a) as a subgraph (where Ω, Δ₁, and Δ₂ are maximal complete subgraphs of Γ) has condition C.(b) Suppose 𝑛 is odd and Γ has maximal complete subgraphs Ω,Δ₁,…,Δ_𝑛. If the vertex set of Ω is 𝑍={𝑣₁,…,𝑣_𝑛} and the vertex set of Δ_𝑖 contains every element of 𝑍 but 𝑣_𝑖, then Γ has condition C. For example, any Coxeter diagram Γ that contains Figure 1(b) as a subgraph (where Ω, Δ₁,Δ₂, and Δ₃ are maximal complete subgraphs of Γ) has condition C.

(a)

(b)

Figure 1 

Remark 5.3. In [5], Tits says that a right-angled Coxeter group with Coxeter diagram Γ has “propriété I” if the complementary graph Γ^𝐶 has no triangles. In this case, the inclusion Inn(𝑊)→Aut^∘(𝑊) is an isomorphism. If Γ has condition C, then the union Δ₁∪⋯∪Δ_𝑛∪Ω defines a right-angled Coxeter group that has propriété I. Thus, condition C is in some sense a “local” version of Tits' propriété I.

Assume Γ satisfies condition C and let 𝜙∈Fix^∘(𝑍). In this case, we can choose 𝑎_𝑍(𝜙)=1. Then, for each 𝑇∈𝔐(Γ) with 𝑇∩𝑍≠∅, the transition 𝛾_𝑍𝑇(𝜙)=𝑎_𝑇(𝜙) is an element of the coset 𝑎_𝑇𝑊_𝑇. It must be emphasized that no left multiplication by elements of 𝑊_𝑍 is permitted. Theorem 5.4. If Γ has condition C, then the sequence1→Inn(𝑊)→Aut^∘(𝑊)→Out^∘(𝑊)→1 (5.1)splits.

Proof. Let 𝑍,𝑇₁,…,𝑇_𝑛∈𝔐(Γ) satisfying (C1) and (C2) of Definition 5.1. For each 𝜙∈Fix^∘(𝑍), define an inner automorphism 𝑟(𝜙)=𝜓_𝑥 where𝑥=^𝑛_𝑖=1𝜋_𝑇𝑖(𝛾_𝑍𝑇𝑖(𝜙))=^𝑛_𝑖=1𝜋_𝑇𝑖(𝑎_𝑇𝑖(𝜙)) (5.2)(recall that 𝜓_𝑥 is the inner automorphism that conjugates each element of 𝑊 by 𝑥). By definition, 𝜋_𝑇𝑖(𝛾_𝑍𝑇𝑖(𝜙))=𝜋_𝑇𝑖(𝑎_𝑇𝑖(𝜙)) lies in 𝑊_{𝑍−𝑇𝑖}⊆𝑊_𝑍. Consequently, the terms in the product (5.2) commute and so the mapping𝑟∶Fix^∘(𝑍)→Inn(𝑍) (5.3)is well defined. To see that 𝑟 is a homomorphism, choose 𝜙,𝜅∈Fix^∘(𝑍). Then, for each 𝑇_𝑖 we have𝑎_𝑇𝑖(𝜙𝜅)=𝜙(𝑎_𝑇𝑖(𝜅))⋅𝑎_𝑇𝑖(𝜙). (5.4)Since 𝑎_𝑇𝑖(𝜅) lies in the centralizer of 𝑊_{𝑇𝑖∩𝑍}, any reduced expression for 𝑎_𝑇𝑖(𝜅) is of the form 𝑎_𝑇𝑖(𝜅)=𝑣₁⋯𝑣_𝑚, where each 𝑣_𝑗∈𝐶(𝑇_𝑖∩𝑍). Observe that𝜋_𝑇𝑖(𝜙(𝑣_𝑗𝑣))=^𝑗,if𝑣_𝑗∈𝑍−𝑇_𝑖,1,if𝑣_𝑗∉𝑍−𝑇_𝑖,=𝜋_𝑇𝑖(𝑣_𝑗) (5.5)and so 𝜋_𝑇𝑖(𝜙(𝑎_𝑇𝑖(𝜅)))=𝜋_𝑇𝑖(𝑎_𝑇𝑖(𝜅)). It follows from (5.4) that𝜋_𝑇𝑖(𝑎_𝑇𝑖(𝜙𝜅))=𝜋_𝑇𝑖(𝑎_𝑇𝑖(𝜅))⋅𝜋_𝑇𝑖(𝑎_𝑇𝑖(𝜙)). (5.6)Now, using the fact that the image of each 𝜋_𝑇𝑖 lies in the Abelian subgroup 𝑊_𝑍, we have that 𝑟(𝜙𝜅)=𝑟(𝜙)𝑟(𝜅).
To see that 𝑟 is a retraction, let 𝑣∈𝑍 and let 𝑘 be a number of 𝑇_𝑖's that do not contain 𝑣. If 𝜓_𝑣 is the inner automorphism of 𝑊 that conjugates by 𝑣, then 𝑎_𝑇𝑖(𝜓_𝑣)=𝑣 for every 1≤𝑖≤𝑛. From (5.2) we have that 𝑟(𝜓_𝑣) conjugates every element by^𝑛_𝑖=1𝜋_𝑇𝑖(𝑎_𝑇𝑖(𝜓_𝑣))=^𝑛_𝑖=1𝜋_𝑇𝑖(𝑣)=𝑣^𝑘. (5.7)Since 𝑘 is odd, 𝑣^𝑘=𝑣 and so 𝑟(𝜓_𝑣)=𝜓_𝑣.
Since 𝑟 is a retraction and the sequence (4.3) is central, the mappingFix^∘(𝑍)→Inn(𝑍)×ker(𝑟) (5.8)defined by𝜙↦(𝑟(𝜙),𝑟(𝜙)⁻¹𝜙) (5.9)is an isomorphism. Consequently, 𝑗=𝜋|_ker(𝑟) is an isomorphism from ker(𝑟) onto Out^∘(𝑊). A section Out^∘(𝑊)→Aut^∘(𝑊) is given by the composition ℎ∘𝑗⁻¹, where ℎ∶ker(𝑟)→Aut^∘(𝑊) is the inclusionker(𝑟)⊆Fix^∘(𝑍)⊆Aut^∘(𝑊). (5.10)

As noted in the introduction, to obtain a splitting of sequence (1.2), Γ must satisfy an “asymmetry” condition. This is obtained by imposing a restriction on the action of Aut(𝔉(Γ)), and motivates the following definition. Definition 5.5. Let 𝛼∈Aut(𝔉(Γ)) and 𝑇∈𝔐(Γ). One says that 𝛼fixes𝑇 if 𝛼({𝑡})={𝑡} for every 𝑡∈𝑇.

Note that if𝛼 fixes𝑇, then𝛼(𝑇^′)=𝑇^′ for every𝑇^′⊆𝑇.Theorem 5.6. If Γ has condition C and each 𝛼∈Aut(𝔉(Γ)) fixes 𝑍, then1→Inn(𝑊)→Aut(𝑊)→Out(𝑊)→1 (5.11)splits.

Proof. Let Fix(𝑍)={𝛽∈Aut(𝑊)∣𝛽(𝑧)=𝑧forall𝑧∈𝑍}. Since each 𝛼∈Aut(𝔉(Γ)) fixes 𝑍, it follows that 𝛼 (as defined in (3.3) above) lies in Fix(𝑍) and the mapping 𝛼↦𝛼 is a splitting of the sequence1→Fix^∘(𝑍)→Fix(𝑍)−−−^𝑞→Aut(𝔉(Γ))→1 (5.12)(where 𝑞 is the restriction of the mapping 𝑞 given in (3.2) above). As in the proof of Theorem 4.4, the projection Fix(𝑍)→Out(𝑊) is onto and its kernel isInn(𝑍)∩Fix(𝑍)=Inn(𝐶(𝑍)). (5.13)But 𝑍∈𝔐(Γ) and so 𝐶(𝑍)=𝑍. Thus, we have an extension0→Inn(𝑍)→Fix(𝑍)→Out(𝑊)→1 (5.14)which is easily seen to be central.
If 𝜙∈Fix^∘(𝑍) and 𝛼∈Aut(𝔉(Γ)), then 𝛼𝜙𝛼⁻¹∈Fix^∘(𝑍) and, because Inn(𝑍) is Abelian,𝑟(𝛼𝜙𝛼⁻¹)=𝑟(𝜙). (5.15)Since (5.12) splits, every element 𝑓∈Fix(𝑍) has a unique expression 𝑓=𝜙⋅𝛼, where 𝜙∈Fix^∘(𝑍) and 𝛼∈Aut(𝔉(Γ)). Consequently, we may define𝑟^′∶Fix(𝑍)→Inn(𝑍) (5.16)by 𝑟^′(𝑓)=𝑟(𝜙) for all 𝑓∈Fix(𝑍). If 𝑔∈Fix(𝑍) is written as 𝜓⋅𝛽, then, by (5.15),𝑟^′(𝑓𝑔)=𝑟^′(𝜙𝛼𝜓𝛼⁻¹𝛼𝛼𝛽)=𝑟(𝜙𝛼𝜓⁻¹)=𝑟(𝜙)𝑟(𝛼𝜓𝛼⁻¹)=𝑟(𝜙)𝑟(𝜓)=𝑟^′(𝑓)𝑟^′(𝑔). (5.17)Thus, 𝑟^′ is a homomorphism and is easily verified to be a retraction. The proof is completed in a manner similar to the conclusion of the proof of Theorem 5.4, replacing Out^∘(𝑊) with Out(𝑊), and so on.

As an immediate consequence we obtain the following corollary (cf. [6]). Corollary 5.7. If 𝑊 is a right-angled Coxeter group satisfying the hypotheses of Theorem 5.6, then every extension1→𝑊→𝐺→𝑄→1 (5.18)splits.Example 5.8. The graph in Figure 2 clearly satisfies the hypothesis of Theorem 5.6. On the other hand, Theorem 5.6 does not apply to either of the graphs given in Figure 1.

Figure 2 

Remark 5.9. As noted in the introduction, it is currently unknown whether, for 𝑊 with trivial center, the sequence1→Inn(𝑊)→Aut(𝑊)→Out(𝑊)→1 (5.19)always splits. However, it was recently shown in [8] that sequence (1.5) always splits, but the splitting found there is not, in general, compatible with (1.1). In particular, one cannot obtain generalizations of Theorem 5.6 and Corollary 5.7 from [8].